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G = C2×C22.26C24  order 128 = 27

Direct product of C2 and C22.26C24

direct product, p-group, metabelian, nilpotent (class 2), monomial

Aliases: C2×C22.26C24, C23.15C24, C22.31C25, C24.477C23, C42.739C23, (C22×C4)⋊50D4, (C4×D4)⋊91C22, (C2×C4).34C24, C4⋊Q8⋊105C22, C2.12(D4×C23), C4⋊D4⋊96C22, C4⋊1D4⋊57C22, C4⋊C4.458C23, (C22×C42)⋊24C2, (C2×C42)⋊92C22, C4.206(C22×D4), C23.408(C2×D4), C22.1(C22×D4), (C2×D4).445C23, C4.4D4⋊96C22, C22⋊C4.72C23, (C2×Q8).418C23, C4○(C22.26C24), (C23×C4).581C22, (C22×C4).1176C23, (C22×D4).584C22, (C22×Q8).486C22, C4○(C2×C4⋊Q8), (C2×C4×D4)⋊71C2, C4○(C2×C4⋊1D4), C4⋊1(C2×C4○D4), C4○3(C2×C4⋊D4), (C2×C4)⋊16(C2×D4), (C2×C4)○2(C4⋊Q8), (C2×C4⋊Q8)⋊62C2, C4○2(C2×C4.4D4), (C2×C4)⋊14(C4○D4), (C2×C4)○2(C4⋊1D4), (C2×C4)○4(C4⋊D4), (C2×C4⋊1D4)⋊30C2, (C2×C4⋊D4)⋊77C2, (C2×C4)○3(C4.4D4), (C2×C4.4D4)⋊63C2, (C22×C4○D4)⋊12C2, (C2×C4○D4)⋊66C22, (C22×C4)○(C4⋊D4), (C22×C4)○(C4⋊1D4), C2.13(C22×C4○D4), (C2×C4⋊C4).946C22, C22.152(C2×C4○D4), (C2×C22⋊C4).527C22, (C2×C4)○(C22.26C24), (C2×C4)○(C2×C4⋊Q8), (C2×C4)○2(C2×C4⋊1D4), (C2×C4)○2(C2×C4⋊D4), SmallGroup(128,2174)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C22 — C2×C22.26C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C22×C42 — C2×C22.26C24
Lower central
C1 — C22 — C2×C22.26C24
Upper central
C1 — C22×C4 — C2×C22.26C24
Jennings
C1 — C22 — C2×C22.26C24

Generators and relations for C2×C22.26C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=1, f2=c, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede=bd=db, be=eb, bf=fb, bg=gb, fdf-1=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 1324 in 864 conjugacy classes, 452 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C24, C24, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C23×C4, C23×C4, C22×D4, C22×Q8, C2×C4○D4, C2×C4○D4, C22×C42, C2×C4×D4, C2×C4⋊D4, C2×C4.4D4, C2×C4⋊1D4, C2×C4⋊Q8, C22.26C24, C22×C4○D4, C2×C22.26C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, C25, C22.26C24, D4×C23, C22×C4○D4, C2×C22.26C24

Smallest permutation representation of C2×C22.26C24
►On 64 points
Generators in S64
(1 25)(2 26)(3 27)(4 28)(5 37)(6 38)(7 39)(8 40)(9 15)(10 16)(11 13)(12 14)(17 31)(18 32)(19 29)(20 30)(21 35)(22 36)(23 33)(24 34)(41 47)(42 48)(43 45)(44 46)(49 62)(50 63)(51 64)(52 61)(53 59)(54 60)(55 57)(56 58)

(1 11)(2 12)(3 9)(4 10)(5 31)(6 32)(7 29)(8 30)(13 25)(14 26)(15 27)(16 28)(17 37)(18 38)(19 39)(20 40)(21 52)(22 49)(23 50)(24 51)(33 63)(34 64)(35 61)(36 62)(41 53)(42 54)(43 55)(44 56)(45 57)(46 58)(47 59)(48 60)

(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 64)

(1 52)(2 51)(3 50)(4 49)(5 59)(6 58)(7 57)(8 60)(9 23)(10 22)(11 21)(12 24)(13 35)(14 34)(15 33)(16 36)(17 41)(18 44)(19 43)(20 42)(25 61)(26 64)(27 63)(28 62)(29 45)(30 48)(31 47)(32 46)(37 53)(38 56)(39 55)(40 54)

(1 47)(2 48)(3 45)(4 46)(5 52)(6 49)(7 50)(8 51)(9 57)(10 58)(11 59)(12 60)(13 53)(14 54)(15 55)(16 56)(17 35)(18 36)(19 33)(20 34)(21 31)(22 32)(23 29)(24 30)(25 41)(26 42)(27 43)(28 44)(37 61)(38 62)(39 63)(40 64)

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)

(1 31 11 5)(2 32 12 6)(3 29 9 7)(4 30 10 8)(13 37 25 17)(14 38 26 18)(15 39 27 19)(16 40 28 20)(21 59 52 47)(22 60 49 48)(23 57 50 45)(24 58 51 46)(33 55 63 43)(34 56 64 44)(35 53 61 41)(36 54 62 42)
 
G:=sub<Sym(64)| (1,25)(2,26)(3,27)(4,28)(5,37)(6,38)(7,39)(8,40)(9,15)(10,16)(11,13)(12,14)(17,31)(18,32)(19,29)(20,30)(21,35)(22,36)(23,33)(24,34)(41,47)(42,48)(43,45)(44,46)(49,62)(50,63)(51,64)(52,61)(53,59)(54,60)(55,57)(56,58), (1,11)(2,12)(3,9)(4,10)(5,31)(6,32)(7,29)(8,30)(13,25)(14,26)(15,27)(16,28)(17,37)(18,38)(19,39)(20,40)(21,52)(22,49)(23,50)(24,51)(33,63)(34,64)(35,61)(36,62)(41,53)(42,54)(43,55)(44,56)(45,57)(46,58)(47,59)(48,60), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64), (1,52)(2,51)(3,50)(4,49)(5,59)(6,58)(7,57)(8,60)(9,23)(10,22)(11,21)(12,24)(13,35)(14,34)(15,33)(16,36)(17,41)(18,44)(19,43)(20,42)(25,61)(26,64)(27,63)(28,62)(29,45)(30,48)(31,47)(32,46)(37,53)(38,56)(39,55)(40,54), (1,47)(2,48)(3,45)(4,46)(5,52)(6,49)(7,50)(8,51)(9,57)(10,58)(11,59)(12,60)(13,53)(14,54)(15,55)(16,56)(17,35)(18,36)(19,33)(20,34)(21,31)(22,32)(23,29)(24,30)(25,41)(26,42)(27,43)(28,44)(37,61)(38,62)(39,63)(40,64), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,31,11,5)(2,32,12,6)(3,29,9,7)(4,30,10,8)(13,37,25,17)(14,38,26,18)(15,39,27,19)(16,40,28,20)(21,59,52,47)(22,60,49,48)(23,57,50,45)(24,58,51,46)(33,55,63,43)(34,56,64,44)(35,53,61,41)(36,54,62,42)>;
 
G:=Group( (1,25)(2,26)(3,27)(4,28)(5,37)(6,38)(7,39)(8,40)(9,15)(10,16)(11,13)(12,14)(17,31)(18,32)(19,29)(20,30)(21,35)(22,36)(23,33)(24,34)(41,47)(42,48)(43,45)(44,46)(49,62)(50,63)(51,64)(52,61)(53,59)(54,60)(55,57)(56,58), (1,11)(2,12)(3,9)(4,10)(5,31)(6,32)(7,29)(8,30)(13,25)(14,26)(15,27)(16,28)(17,37)(18,38)(19,39)(20,40)(21,52)(22,49)(23,50)(24,51)(33,63)(34,64)(35,61)(36,62)(41,53)(42,54)(43,55)(44,56)(45,57)(46,58)(47,59)(48,60), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64), (1,52)(2,51)(3,50)(4,49)(5,59)(6,58)(7,57)(8,60)(9,23)(10,22)(11,21)(12,24)(13,35)(14,34)(15,33)(16,36)(17,41)(18,44)(19,43)(20,42)(25,61)(26,64)(27,63)(28,62)(29,45)(30,48)(31,47)(32,46)(37,53)(38,56)(39,55)(40,54), (1,47)(2,48)(3,45)(4,46)(5,52)(6,49)(7,50)(8,51)(9,57)(10,58)(11,59)(12,60)(13,53)(14,54)(15,55)(16,56)(17,35)(18,36)(19,33)(20,34)(21,31)(22,32)(23,29)(24,30)(25,41)(26,42)(27,43)(28,44)(37,61)(38,62)(39,63)(40,64), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,31,11,5)(2,32,12,6)(3,29,9,7)(4,30,10,8)(13,37,25,17)(14,38,26,18)(15,39,27,19)(16,40,28,20)(21,59,52,47)(22,60,49,48)(23,57,50,45)(24,58,51,46)(33,55,63,43)(34,56,64,44)(35,53,61,41)(36,54,62,42) );
 
G=PermutationGroup([[(1,25),(2,26),(3,27),(4,28),(5,37),(6,38),(7,39),(8,40),(9,15),(10,16),(11,13),(12,14),(17,31),(18,32),(19,29),(20,30),(21,35),(22,36),(23,33),(24,34),(41,47),(42,48),(43,45),(44,46),(49,62),(50,63),(51,64),(52,61),(53,59),(54,60),(55,57),(56,58)], [(1,11),(2,12),(3,9),(4,10),(5,31),(6,32),(7,29),(8,30),(13,25),(14,26),(15,27),(16,28),(17,37),(18,38),(19,39),(20,40),(21,52),(22,49),(23,50),(24,51),(33,63),(34,64),(35,61),(36,62),(41,53),(42,54),(43,55),(44,56),(45,57),(46,58),(47,59),(48,60)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32),(33,35),(34,36),(37,39),(38,40),(41,43),(42,44),(45,47),(46,48),(49,51),(50,52),(53,55),(54,56),(57,59),(58,60),(61,63),(62,64)], [(1,52),(2,51),(3,50),(4,49),(5,59),(6,58),(7,57),(8,60),(9,23),(10,22),(11,21),(12,24),(13,35),(14,34),(15,33),(16,36),(17,41),(18,44),(19,43),(20,42),(25,61),(26,64),(27,63),(28,62),(29,45),(30,48),(31,47),(32,46),(37,53),(38,56),(39,55),(40,54)], [(1,47),(2,48),(3,45),(4,46),(5,52),(6,49),(7,50),(8,51),(9,57),(10,58),(11,59),(12,60),(13,53),(14,54),(15,55),(16,56),(17,35),(18,36),(19,33),(20,34),(21,31),(22,32),(23,29),(24,30),(25,41),(26,42),(27,43),(28,44),(37,61),(38,62),(39,63),(40,64)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64)], [(1,31,11,5),(2,32,12,6),(3,29,9,7),(4,30,10,8),(13,37,25,17),(14,38,26,18),(15,39,27,19),(16,40,28,20),(21,59,52,47),(22,60,49,48),(23,57,50,45),(24,58,51,46),(33,55,63,43),(34,56,64,44),(35,53,61,41),(36,54,62,42)]])
 

56 conjugacy classes

class 1 2A···2G2H2I2J2K2L···2S4A···4H4I···4AB4AC···4AJ
order12···222222···24···44···44···4
size11···122224···41···12···24···4

56 irreducible representations

dim11111111122
type++++++++++
imageC1C2C2C2C2C2C2C2C2D4C4○D4
kernelC2×C22.26C24C22×C42C2×C4×D4C2×C4⋊D4C2×C4.4D4C2×C4⋊1D4C2×C4⋊Q8C22.26C24C22×C4○D4C22×C4C2×C4
# reps1144211162816

Matrix representation of C2×C22.26C24 ►in GL5(𝔽5)

40000
04000
00400
00010
00001
,
10000
04000
00400
00010
00001
,
10000
01000
00100
00040
00004
,
40000
01400
00400
00040
00011
,
40000
02300
04300
00010
00001
,
10000
04000
00400
00012
00044
,
40000
03000
00300
00040
00004

G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,1,0,0,0,0,4,4,0,0,0,0,0,4,1,0,0,0,0,1],[4,0,0,0,0,0,2,4,0,0,0,3,3,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,4,0,0,0,2,4],[4,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,4,0,0,0,0,0,4] >;
 

C2×C22.26C24 in GAP, Magma, Sage, TeX 

C_2\times C_2^2._{26}C_2^4
 
% in TeX
 
G:=Group("C2xC2^2.26C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,2174);
 
// by ID
 
G=gap.SmallGroup(128,2174);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,352,136]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=1,f^2=c,g^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,e*d*e=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,f*d*f^-1=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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